Rigidity Theorems for Multiparametric Deformations of the Universal Enveloping Algebras

Abstract

The well-known one-dimensional Riemann–Hilbert problem on C concerns the proof of the existence of the Fuchsian system on C with a given monodromy, that is, with a given representation of a fundamental group, which is a braid group. In higher dimensions, the classical statement of the problem is modified so that instead of proving the existence of a system of differential equations of Fuchsian type, the problem of characterization of the representations of a fundamental group that can be a monodromy representation of the differential system of Fuchsian type is stated. It is a purely algebraic problem, but since it is closely connected with the question concerning the recovery of the functions of several complex variables if their ramification is given, its solution is based on analytical methods connected with properties of the power series on the parameters of the differential equation. In the context of the Riemann–Hilbert problem, to these methods should be added some algebraic ones concerning the algebraic structures that can be defined on algebras of such series. Some of these methods are considered in this paper, in particular, the method of multi-parametric deformations of some algebras, more exactly, the quantum enveloping algebras. In the one-parametric case, the quantum enveloping algebras are the algebras over the ring of the formal power series C[[h]]. The last object carries the topological structure defined by the powers of an ideal generated by a (quantum) parameter h. This topological structure naturally induces some structures on algebras and on modules over algebras. All methods of proving the different statements about these structures naturally are based on induction on the powers of h. Obviously, such a technique can be extended to the formal power series of several variables C[[h1, . . . , hn]]. Since many of the proofs of the theorems are easily carried over from the one-parametric case to the multi-parametric one, we have hopes that a variety of facts from the classical theory of quantum groups are valid for multi-parametric deformations. This paper is devoted to problems connected with multi-parametric generalizations of methods that are important for the multi-parametric analogue of the Drinfeld–Kohno theory. The well-known Drinfeld–Kohno theorem is a statement about the conditions of solvability of the Riemann–Hilbert problem in a class of the Knizhnik–Zamolodchikov equations [2, 7], which are particular cases of the multidimensional Fuchsian systems. Their proof and some of their generalizations can be found in [2,3,5,7]. In this paper, we first introduce some definitions that are used in the extensions of the Drinfeld– Kohno theory: the definition of the multi-parametric Drinfeld–Jimbo algebra and definitions of the braided multi-parametric quantum enveloping algebra and the braided multi-parametric quasi-bialgebra. We introduce the isomorphism of the quasi-bialgebras up to the twisting. Further, the uniqueness theorems for the homomorphisms of the trivial multi-parametric deformation of the enveloping algebra if these homomorphisms coincide on the universal enveloping algebras are proved. Then we prove some theorems on the uniqueness for the quantum enveloping algebras and for the Drinfeld–Jimbo algebra. We also prove the theorem on the isomorphism (up to the twisting) of the braided quasi-bialgebras having different associators. All these results are the necessary components in a proof of the multi-parametric analogue of the Drinfeld theorem on the isomorphism of the structure of a braided quasi-bialgebra (on